z score Calculator

The probability of a result x in an experiment consisting of a
large number of equally probable independent trials n is
approximated by the normal probability density function:

where μ, the mean value, is n/2 and
σ,
the standard deviation, is a measure of the breadth of the
curve which, for experiments with two equally probable outcomes
of each trial is:

Since the mean value and standard deviation depend upon the number
of trials in the experiment, comparison between experiments
with differing numbers of trials is facilitated by
standardising the result: transforming it
to a distribution with mean value zero and standard deviation
of 1. A normally distributed experimental result x
is thus standardised by subtracting the mean and dividing
by the standard deviation of the experiment:

This z-value or z score expresses the
divergence of the experimental result x from the
most probable result μ as a number of standard
deviations σ
The larger the value of z, the less probable the
experimental result is due to chance. The probability
can be calculated from the cumulative standard
normal distribution:

Which gives the probability P that an experimental
result with a z value less than or equal to that
observed is due to chance. Subtracting P from one:

gives Q, the probability that the observed z
score is due to chance.

Unfortunately, there is no closed form solution for P, so it
must be evaluated numerically. This page allows you to calculate the
probability of chance occurrence of a given z, or to
calculate z given a probability Q. All
calculations are performed with six decimal places of accuracy; the
maximum z accepted is 6.

Note that the probability calculated from the z
is an approximation which is valid only for large
values of n, and is therefore only meaningful when
calculated for experiments with a large number of
individual trials.

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Calculate probability Q from z

One of the most common probability calculations is determining, given
the measured z value from an experiment or set of
experiments, the probability of the result being due to chance. Enter
the z value in the box below, press the Return key or
the Calculate button, and the probability will appear in the Q
box.

Given z =

The chance probability, Q, is:
or about one in

Calculate z from probability Q

To determine the z score indicating a probability Q
of non-chance occurrence for an experiment,
enter Q in the box below and
press the Return key or the Calculate button.